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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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10 Diagnostic Tests 630 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #36 : Factorials

Divide the factorials: $\frac{8!}{(2+4)!}$

Possible Answers:

$\textup{The answer is not given.}$

Correct answer:

Explanation:

Simplify the denominator in the parentheses first.

$\frac{8!}{(2+4)!} = \frac{8!}{6!}$

Rewrite the factorials.

$\frac{8!}{6!} = \frac{8\times7\times6\times5\times4\times3\times2\times1 }{6\times5\times4\times3\times2\times1}$

Cancel all the common terms in the numerator and denominator.

The numerator becomes: $8\times 7 =56$

The answer is:

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Example Question #37 : Factorials

Multiply the following factorials: $(1+2)! \cdot 3!$

Possible Answers:

Correct answer:

Explanation:

In order to simplify the factorials, first simplify the parentheses.

$(1+2)! \cdot 3!= 3! \cdot 3!$

Simplify the factorials.

$(3\times 2 \times 1)\cdot(3\times 2 \times 1) = (6) \cdot (6) =36$

The answer is:

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Example Question #38 : Factorials

Multiply the factorials: $\frac{(1+0!)!}{0!+3!}$

Possible Answers:

$\frac{1}{3}$

$\frac{1}{12}$

$\frac{2}{7}$

$\frac{1}{6}$

$\frac{1}{4}$

Correct answer:

$\frac{2}{7}$

Explanation:

The value of zero factorial is one.

0!=1

Replace this value.

$\frac{(1+0!)!}{0!+3!}=\frac{(1+1!)!}{1+3!}$

Expand the factorials.

$\frac{(1+1!)!}{1+3!}=\frac{(1+1)!}{1+(3\times 2\times 1)}= \frac{2}{1+6}=\frac{2}{7}$

The answer is: $\frac{2}{7}$

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Example Question #31 : Multiplying And Dividing Factorials

Divide the factorials: $\frac{(3+5)!}{6!}$

Possible Answers:

$\frac{7}{40}$

$\frac{4}{3}$

Correct answer:

Explanation:

Simplify the numerator of the fraction.

$\frac{(3+5)!}{6!} = \frac{8!}{6!}$

Do not reduce the fraction and cancel out the factorials. Instead, expand both.

$\frac{8!}{6!} = \frac{8\times7\times6\times5\times4\times3\times2\times1}{6\times5\times4\times3\times2\times1}$

Notice that the similar terms on the numerator and denominator will cancel.

The remaining terms are: $8\times 7= 56$

The answer is:

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Example Question #40 : Factorials

Divide the factorials: $\frac{4!(4-2)!}{(1+2)!}$

Possible Answers:

120

6720

2240

Correct answer:

Explanation:

Simplify every term inside the parentheses.

$\frac{4!(4-2)!}{(1+2)!} = \frac{4!(2)!}{(3)!}$

Expand each factorial.

$\frac{(4\times 3\times 2\times 1)(2\times 1)}{3\times 2\times 1}$

Cancel out the common terms on the numerator and denominator and rewrite what remains of the numerator.

$4\times 2\times 1=8$

The answer is:

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Example Question #31 : Multiplying And Dividing Factorials

Divide the factorials. $\frac{3! \cdot5!}{4!\cdot6!}$

Possible Answers:

$\frac{1}{24}$

$\frac{1}{36}$

$\frac{5}{8}$

$\frac{1}{48}$

$\frac{1}{72}$

Correct answer:

$\frac{1}{24}$

Explanation:

Expand each factorial.

$\frac{3! \cdot5!}{4!\cdot6!}=\frac{(3\times2\times1) (5\times4 \times3\times2\times1)}{ (4 \times3\times2\times1)(6\times 5\times4 \times3\times2\times1)}$

Cancel out the common terms on the numerator and denominator.

The remain terms are:

$\frac{1}{4(6)} = \frac{1}{24}$

The answer is: $\frac{1}{24}$

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Example Question #42 : Factorials

Divide the fractions: $\frac{3}{8}\div\frac{9}{34}$

Possible Answers:

$\frac{12}{17}$

$\frac{18}{23}$

$\frac{272}{27}$

$\frac{17}{12}$

$\frac{23}{18}$

Correct answer:

$\frac{17}{12}$

Explanation:

In order to divide these fractions, we will need to take the reciprocal of the second term and change the sign to a multiplication.

$\frac{3}{8}\times \frac{34}{9}$

Write each non-prime term in factors.

$\frac{3}{2\times 4}\times \frac{2\times 17}{3 \times 3}$

Reduce the terms by cancelling the common factors in the numerator and denominator.

$\frac{17}{4\times 3} = \frac{17}{12}$

The answer is: $\frac{17}{12}$

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Example Question #43 : Factorials

Multiply the factorials: $0!\cdot 2! \cdot 4!$

Possible Answers:

720

Correct answer:

Explanation:

Identify the values of each factorial. Zero factorial is a special case.

0!=1

$2! = 2\times 1 = 2$

$4! =4\times 3 \times 2\times 1=24$

Multiply all three numbers.

$0!\cdot 2! \cdot 4!=1\cdot2\cdot24 = 48$

The answer is:

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Example Question #44 : Factorials

Simplify the factorials: $\frac{5(3+2)!}{5!\cdot 3!}$

Possible Answers:

$\frac{1}{2}$

$1.186\times 10^{13}$

$\frac{3}{4}$

$\frac{5}{3}$

$\frac{5}{6}$

Correct answer:

$\frac{5}{6}$

Explanation:

Simplify the numerator. Do not distribute the five through the terms inside the parentheses, or it will change the value of the factorial!

$5(3+2)! = 5(5)! = 5[5\cdot 4\cdot 3\cdot 2\cdot 1]$

Simplify the denominator.

$5!\cdot 3! = [5\cdot 4\cdot 3\cdot 2\cdot 1]\cdot [3\cdot 2\cdot 1]$

Divide the numerator with the denominator.

$\frac{5[5\cdot 4\cdot 3\cdot 2\cdot 1]}{[5\cdot 4\cdot 3\cdot 2\cdot 1]\cdot [3\cdot 2\cdot 1]}= \frac{5}{3\cdot 2\cdot 1}= \frac{5}{6}$

The answer is: $\frac{5}{6}$

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Example Question #45 : Factorials

Simplify: $\frac{4(1+3)!}{2(3)!}$

Possible Answers:

$2.91\times 10^{10}$

$1.74\times 10^{11}$

6720

Correct answer:

Explanation:

Simplify the terms in side the parentheses first. Do NOT distribute the four through the binomial or this will change the value of the factorial!

$\frac{4(1+3)!}{2(3)!} = \frac{4(4)!}{2(3\cdot 2\cdot 1)} = \frac{4(4\cdot 3\cdot 2\cdot 1)}{2(3\cdot 2\cdot 1)}$

Notice the there are common terms in the numerator and denominator. Cancel out the terms. The remaining parts are:

$\frac{4\cdot 4}{2} = 8$

The answer is:

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #36 : Factorials

Example Question #37 : Factorials

Example Question #38 : Factorials

Example Question #31 : Multiplying And Dividing Factorials

Example Question #40 : Factorials

Example Question #31 : Multiplying And Dividing Factorials

Example Question #42 : Factorials

Example Question #43 : Factorials

Example Question #44 : Factorials

Example Question #45 : Factorials

All Algebra II Resources

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